> On 21 Mrz., 09:14, Virgil <vir...@ligriv.com> wrote:> > While induction can certainly prove that no finite set of lines is> > sufficient, it cannot prove that an infinite set of lines cannot be> > sufficient,> > It can prove that no finite line and no set of finite lines is> sufficient. Induction holds for every line of the list.

How does induction prove the set of all finite lines is not sufficient to contain all lines and thus all natualsr?

Do show us! In detail!

> > > because some infinite sets of lines clearly ARE sufficient,> > the set of all lines, for example, is both infinite and sufficient.> > Ah so, "they clearly are".> Contemplate this parallel claim:> An infinite set of natural numbers clearly contains an infinite> number. Therefore infinite set of natural numbers clearly are> containing infinite numbers.

That claim is not parallel to anything outside Wolkenmuekenheim.

And within Wolkenmuekenheim a pair of parallel lines may well also be perpendicular to each other, since all sorts of impossible things are possible there, at least according to WM.

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WM claims to know how to map bijectively the set of infinite binary sequences, B, linearly to the set of reals and then map that image set of reals linearly ONTO the set of all paths, P, of a Complete Infinite Binary Tree.

But each binary rational in |R is necessarily the image of two sequences in B but that one rational can then only produce one image in P, so the mapping cannot be the bijection WM claims.

So that WM is, as usual with things mathematical, claiming impossibilities.--